The present invention provides an air-oxy-fuel method and apparatus in which fuel is burned in first and second stages of combustion having a fuel-rich and fuel-lean stoichiometry, respectively, to reduce NO.sub.x, emissions. The fuel is burned in the first stage of combustion in oxygen. The combustion of the fuel is completed in the second stage of combustion in the presence of air. Mixing of fuel and oxygen is effected by forming fuel and oxygen jets in close proximity to one another. Air is then swirled around the fuel and oxygen jets at a swirl ratio of no less than about 0.6 so that combustion products of the fast stage of combustion recirculate to obtain stable combustion of a very fuel-rich flame which is characterized by increased flame luminosity. The use of air allows for a lesser degree of swirling than had a single oxidant been used in both the fast and second stages of combustion because a greater mass of air is supplied than oxygen. This in turn allows the air to be swirled with a sufficient forward momentum to locate the second stage of combustion directly downstream of the fast stage of combustion.